HUNGARIAN JOURNAL OF INDUSTRY AND CHEMISTRY VESZPRÉM Vol. 42(2) pp. 109 113 (2014) STABILITY AND PARAMETER SENSITIVITY ANALYSES OF AN INDUCTION MOTOR ATTILA FODOR 1, ROLAND BÁLINT 1, ATTILA MAGYAR 1, AND GÁBOR SZEDERKÉNYI 2 1 Depatment of Electical Engineeing and Infomation Systems, Univesity of Pannonia, Egyetem u. 10., Veszpém, 8200, HUNGARY 2 Pazmány Péte Catholic Univesity, Páte u. 50/a, Budapest, 1083, HUNGARY E-mail: balint.oland27@gmail.com A simple dynamical model of an induction moto is deived and analyzed in this pape based on engineeing pinciples that descibe the mechanical phenomena togethe with the electical model. The used state space model consists of nonlinea state equations. The model has been veified unde the usual contolled opeating conditions when the speed is contolled. The effect of load on the contolled induction moto has been analyzed by simulation. The sensitivity analysis of the induction moto has been applied to detemine the model paametes to be estimated. Keywods: induction moto, stability analysis, sensitivity analysis Intoduction Induction motos (IM) ae the most commonly used electical otating machines in seveal industial applications. Iespective of size and the application aea, these motos shae the most impotant dynamical popeties, and thei dynamical models have a simila stuctue. Because of the specialties and geat pactical impotance of IMs in industial applications, thei modelling fo contol puposes is well investigated in the liteatue. Besides basic textbooks [1-3], thee ae seveal papes that descibe the modelling and use the developed models fo the design of diffeent types of contolles: vecto contol [1, 4], sensoless vecto contol [5] and diect toque contol (DTC) [6]. The aim of this pape is to build a simple dynamical model of the IM and to pefom its paamete sensitivity analysis. The esults of this analysis will be the basis of the next step since the final aim of ou study is to estimate the paametes of the IM and design a contolle that can contol the speed and toque of the IM. The state space model has been implemented in the Matlab/Simulink envionment which enables us to analyze the paametic sensitivity based on simulation expeiments. Nonlinea Model of an Induction Moto In this section a state space model of an induction moto (IM) is pesented. The model development is lagely based on Refs.[1, 8-10]. Fo constucting the IM model, the following modelling assumptions ae made: 1. a symmetical tiphase stato winding system is assumed, 2. the flux density is adial in the ai gap, 3. the coppe loss and the slots in the machine can be neglected, 4. the spatial distibution of the stato fluxes and apetues wave ae consideed to be sinusoidal, 5. stato and oto pemeability ae assumed to be infinite with linea magnetic popeties. Accoding to the above modelling conditions the mathematical desciption of the IM is developed though the space-vecto theoy. If the voltage of the stato is pesumed to be the input excitation of the machine, then the spatial distibution along the stato of the x phase voltage can be descibed by the complex vecto v sx (t). We can detemine the oientation of the voltage vecto v s, the diection of the espective phase axis and the voltage polaity. i s (t) 2 ( a0 i sa (t) + a 1 i sb (t) + a 2 i sc (t) ) 3 2 i eff (t) e jω0t+ π 2 +ϕi, (1) whee a is the e j120o vecto and i sa, i sb and i sc ae the following: i sa (t) Re(a 0 i s (t)) Re(i s (t)), i sb (t) Re(a 2 i s (t)), and i sc (t) Re(a 1 i s (t)). In Eq.(1), 2/3 is the nomalizing facto. The flux density distibution can be obtained by integating the cuent density wave along the cylinde of the stato. The flux linkage wave is a system vaiable, because it contains detailed infomation about the winding geomety.
110 Figue 1: The equivalent cicuit of the IM The otating flux density wave induces voltages in the individual stato windings. Thus stato voltage v s (t) can be epesented as the oveall distibuted voltages in all phase windings: v s (t) 2 ( a0 v sa (t) + a 1 v sb (t) + a 2 v sc (t) ) 3 2 v eff (t) e jω0t+ π 2 +ϕu, (2) whee a is the e j120o vecto and i sa, i sb and i sc ae the following: v sa (t) Re(a 0 u s (t)) Re(u s (t)), v sb (t) Re(a 2 u s (t)), and v sc (t) Re(a 1 u s (t)). Consideing the stato of the IM as the pime side of the tansfome, then using Kichoff s voltage law the following equation can be witten (Fig.1): v s (t) i s (t) R s + dφ s(t). (3) As fo the seconday side of the tansfome, it can be deduced that the same elationship is tue fo the oto side space vectos: v (t) i (t) R + dφ (t) 0. (4) Eqs.(3) and (4) descibe the electomagnetic inteaction as the connection of fist ode dynamical subsystems: φ s (t) i s (t) L s + i (t), and (5) φ (t) i s (t) + i (t) L. (6) Since fou complex vaiables (i s (t), i (t), φ s (t), and φ (t)) ae pesented in Eqs.(5) and (6), flux equations ae needed to complete the elationship between them. The mechanical powe (P mech (t)) of the IM can be defined as: P mech (t) W mech(t), (7) whee the mechanical enegy (P mech (t)) in the otating system can be given by the following expession: P mech (t) W mech(t) T mech (t) ω, (8) whee T e (t) is the toque and ω is the angula velocity of the IM. Aftewads the enegy balance of the IM is as follows: W e W mech + W R + W Field, (9) Figue 2: The equivalent cicuit of the d-axis of the IM Figue 3: The equivalent cicuit of the q-axis of the IM whee P e W e(t) is the input electical powe, P R W R(t) 3 2 Re (u s i s + u i ) (10) 3 2 Re ( R s i s 2 + R i 2) (11) epesents the esistive powe losses and P Field W Field(t) 3 ( 2 Re dφs i s + dφ ) i is the ai gap powe. Using Eqs.(8-12), (12) P mech (t) T mech (t) ω 3 2 Lm φ s (t) i s (t), and L (13) T e 1.5p (φ ds i qs φ qs i ds ). (14) The equivalent cicuit of the IM can be decomposed to diect axis and quadatic axis components by Pak s tansfomation as shown in Figs.2 and 3. The actual teminal voltage v of the windings can be witten in the fom v ± J (R j i j ) ± j1 J j1 ( ) dφj, (15) whee i j ae the cuents, R j ae the winding esistances, and φ j ae the flux linkages. The positive diections of the stato cuents point out of the IM teminals. By composing the d and q axes of the IM the following equations can be witten: v qs R s i qs + φ qs + ω φ ds, (16) v ds R s i ds + φ ds ω φ qs, (17) v q R i ds + φ d + (ω ω ) φ d, and (18) v d R i d + φ d (ω ω ) φ q. (19)
111 dφ d R φ ds + R L s ( ) φ d + ω φ q ω φ q + v d, and (28) dω 1 2H 1 2H 1.5 p (L s L2 m L ) φ d φ qs 1.5 p (L s L2 m L ) φ q φ ds F 2H ω T m 2H. (29) The state vecto of the above model is x[φ qs, φ ds, φ q, φ d, ω ] T R 5, and the input vaiables ae oganized in tems of the the input vecto u[v q, v d, T mech ] T R 3. It is assumed that all the state vaiables can be measued i.e. y x. Model Validation Figue 4: Response to the step change of the speed-contolled IM dω m 1 2H (T e F ω m T mech ) (20) whee ω is the efeence fame angula velocity, ω is the electical angula velocity, φ qs L s i qs + i q, (21) φ ds L s i ds + i d, (22) φ q L i q + i qs, and (23) φ d L i d + i ds. (24) The above model can be witten in state space fom by expessing the time deivative of the fluxes and ω fom the voltage and swing equations Eqs.(21-24). The nonlinea state space model of the IM is given by Eqs.(25-29): dφ qs dφ ds R s R s φ qs L s L2 m L L (L s L2 m L ) φ q ω φ ds + v qs, (25) R s φ ds + R s L s L2 m L L (L s L2 m L ) φ d ω φ qs + v ds, (26) dφ q R φ qs + R L s ( ) φ q ω φ d + ω φ d + v q, (27) The dynamical popeties of the IM have been investigated. The esponse of the speed-contolled moto has been tested unde step-like changes. The simulation esults ae shown in Fig.4, whee the fluxes (φ qs, φ ds, φ q, and φ d ) and the angula velocity (ω) ae shown. Model Analysis The above model Eqs.(21-24) has been veified by simulation against engineeing intuition. Local Stability Analysis As the final aim of ou eseach is to estimate the paametes of a paticula Gundfos IM, fist of all the esistances (R s and R ) of the IM wee measued. Aftewads the values of the inductances (L ls, L l and ) and the mechanical paametes (H and F ) of a simila IM with simila R s and R found in the liteatue have been used. The paametes used duing the model and the sensitivity analyses ae the following: R s 0.196 Ohm R 0.0191 Ohm L ls 0.0397 H L l 0.0397 H 1.354 H H 0.095 F 0.0548 p 1 (30)
112 Figue 5: The model state vaiables fo a ±50% change of It can easily be seen that the IM model is nonlinea since thee ae poducts of two state vaiables in the following equations: Eq.(27): φ d is multiplied by ω Eq.(28): φ q is multiplied by ω Eq.(29): φ d is multiplied by φ qs Eq.(29): φ q is multiplied by φ ds Fo the local stability analysis we have to calculate the eigenvalues of the system. The examined equilibium point is φ qs 0.744 Vs φ ds 1.0287 Vs φ q 0.174 Vs (31) φ d 0.087 Vs ω 48.477 s 1 The numeical value of the Jacobian of the nonlinea model (i.e. the state matix of the locally lineaized model) is as follows: 2.504 50 2.4328 0 0 50 2.504 0 2.4328 0 0.2370 0 0.244 1.522 0 0 0.2370 1.5222 0.244 0 8.617 17.077 0 0 0.288. (32) The eigenvalues of the state matix of the lineaized systems ae: λ 1,2 2.504 ± j49.98, λ 3,4 0.243 ± j1.534, and λ 5 0.288 (33) Figue 6: The model state vaiables fo a ±50% change in R s It is appaent that the eal pats of the eigenvalues ae negative with small magnitudes. Paamete Sensitivity Analysis the sensitivity of the nonlinea model to the mutual inductance has been investigated. The steady state value of the system vaiables does not change (as is appaent in Fig.5) even fo a consideably lage change in. Sensitivity analysis of the inductances L l, L ls of the stato and oto, esistances of the stato R s and the damping constant F has also been investigated. In this investigation it can be seen that the values of the state vaiables wee changed a bit, as shown in Fig.6. The analysis of the esistances of the oto R and the inetia H of the oto showed that evey value of the state vaiables changed significantly, as shown in Fig.7. As a final esult of the sensitivity analysis, we can define the following goups of paametes: Not sensitive: Mutual inductance. Since the state space model of inteest is insensitive in this espect, the values of this paamete cannot be eliably detemined fom measuement data using any paamete estimation method. Sensitive: These sensitive paametes ae candidates fo paamete estimation. Less: inductances L l and L ls, esistance of the stato R s and the damping constant F. Moe: esistances of the oto R and the inetia H of the oto. Citically sensitive: none.
113 SYMBOLS C 1, C 2, C 3 F φ qs and φ ds φ q and φ d H i qs and i ds i q and i d p R and L R and L R s and L s v d and v q ω ω T mech constant damping constant q and d components of the stato flux q and d components of the educed oto flux inetia constant q and d components of the stato cuent q and d components of the educed oto cuent mutual inductance numbe of pole pais oto esistance and inductance educed value of oto esistance and inductance stato esistance and inductance q and d components of the stato voltage angula velocity of the magnetic field angula velocity of the oto mechanical toque Figue 7: The model state vaiables fo a ±50% change in H Conclusion The simple nonlinea dynamical model of an IM has been investigated in this pape. It has been shown that the model is locally asymptotically stable with egads to a physically meaningful equilibium state. The effect of the contolled geneato has been analyzed by simulation using a taditional PI contolle. It has been found that the contolled system is stable and can follow the set-point changes. Seven paametes of the model wee selected fo sensitivity analysis, and the sensitivity of the state vaiables has been investigated. As a esult, the paametes wee patitioned into thee goups. Based on the esults pesented hee, a futue aim is to estimate the paametes of the model fo a eal system fom measuements. The sensitivity analysis enables us to select the candidates fo estimation that ae inductances L l and L ls, esistances R s and R, the damping constant F and the inetia H. An additional futue aim is to develop a model fo contol puposes and investigate diffeent contolles befoe applying them on eal systems. ACKNOWLEDGEMENT We acknowledge the financial suppot of this wok by the Hungaian State and the Euopean Union unde the TAMOP-4.2.2.A-11/1/KONV-2012-0072 poject. We acknowledge also the investigated IM of Gundfos Hungay Poduce Ltd. REFERENCES [1] VAS P.: Atifical-intelligence-Based Electical Machines and Dives, Oxfod Univesity Pess, 1999 [2] VAS P.: Sensoless Vecto and Diect Toque Contol. Oxfod Univesity Pess, 1998. [3] ZHENG L., FLETCHER J.E., WILLIAMS B.W. HE X.: Dual-Plane Vecto Contol of a Five-Phase Induction Machine fo an Impoved Flux Patten, IEEE Tansac. Ind. Elec., 2008, 55(5), 1996 2005 [4] LEVI E.: Impact of Ion Loss on Behavio of Vecto Contolled Induction Machines, IEEE Tansac. Ind. Appl., 1995, 31(6), 1287 1296 [5] HASEGAWA M., MATSUI K.: Robust Adaptive Full-Ode Obseve Design with Novel Adaptive Scheme fo Speed Sensoless Vecto Contolled Induction Motos, Poc. 28th Annual Conf. Industial Electonics Society, IEEE 2002, 2002, 1, 83 88 [6] GEYER T., PAPAFOTIOU G., MORARI M.: Model Pedictive Diect Toque Contol-Pat I: Concept, Algoithm, and Analysis, IEEE Tansac. Ind. Elec., 2009, 56(6), 1894 1905 [7] KRAUSE P.C., WASYNCZUK O., SUDHOFF S.D.: Analysis of Electic Machiney, IEEE Pess, New Yok, 1995 [8] MOHAN N., UNDELAND T.M., ROBBINS W.P.: Powe Electonics: Convetes, Applications, and Design, John Wiley & Sons Inc., New Yok, 1995 [9] HATOS P., FODOR, A., MAGYAR A.: Paamete sensitivity analysis of an induction moto, Hung. J. Ind. Chem., 2011, 39(1), 157 161